The objective of this proposal is to study the transient response of solute-cycling counter current multipliers and to collect and refine the mathematical tools necessary for this study. The study will begin with the simplest models and work toward counter current multipliers that approximate that found in the medulla of a mammalian kidney. This complicated counter flow system is the means by which urine is concentrated and includes the loops of Henle, the collecting ducts, and the vasa recta. Initially, the proposed research will be toward obtaining analytic solutions for the transient response of a number of counter current multipliers. These solutions will then aid in the construction of more complicated systems and will serve as checks for numerical solutions of these systems.